The Annals of Probability

When does a discrete-time random walk in $\mathbb{R}^{n}$ absorb the origin into its convex hull?

Konstantin Tikhomirov and Pierre Youssef

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Abstract

We connect this question to a problem of estimating the probability that the image of certain random matrices does not intersect with a subset of the unit sphere $\mathbb{S}^{n-1}$. In this way, the case of a discretized Brownian motion is related to Gordon’s escape theorem dealing with standard Gaussian matrices. We show that for the random walk $\mathrm{BM}_{n}(i),i\in\mathbb{N}$, the convex hull of the first $C^{n}$ steps (for a sufficiently large universal constant $C$) contains the origin with probability close to one. Moreover, the approach allows us to prove that with high probability the $\pi/2$-covering time of certain random walks on $\mathbb{S}^{n-1}$ is of order $n$. For certain spherical simplices on $\mathbb{S}^{n-1}$, we prove an extension of Gordon’s theorem dealing with a broad class of random matrices; as an application, we show that $C^{n}$ steps are sufficient for the standard walk on $\mathbb{Z}^{n}$ to absorb the origin into its convex hull with a high probability. Finally, we prove that the aforementioned bound is sharp in the following sense: for some universal constant $c>1$, the convex hull of the $n$-dimensional Brownian motion $\operatorname{conv}\{\mathrm{BM}_{n}(t):t\in[1,c^{n}]\}$ does not contain the origin with probability close to one.

Article information

Source
Ann. Probab., Volume 45, Number 2 (2017), 965-1002.

Dates
Received: November 2014
Revised: November 2015
First available in Project Euclid: 31 March 2017

Permanent link to this document
https://projecteuclid.org/euclid.aop/1490947312

Digital Object Identifier
doi:10.1214/15-AOP1079

Mathematical Reviews number (MathSciNet)
MR3630291

Zentralblatt MATH identifier
1377.52008

Subjects
Primary: 52A22: Random convex sets and integral geometry [See also 53C65, 60D05]
Secondary: 60J05: Discrete-time Markov processes on general state spaces 60J65: Brownian motion [See also 58J65]

Keywords
Convex hull random walk covering time

Citation

Tikhomirov, Konstantin; Youssef, Pierre. When does a discrete-time random walk in $\mathbb{R}^{n}$ absorb the origin into its convex hull?. Ann. Probab. 45 (2017), no. 2, 965--1002. doi:10.1214/15-AOP1079. https://projecteuclid.org/euclid.aop/1490947312


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